Friday, 13 July 2012

Ten hours with Erica

This is a true story about how a student in year 6 (Erica is not her real name) who was experiencing very serious difficulties in maths became confident and happy in her work.  I’ve written it down so that other teachers, parents and students can read it and see if there’s anything in it that might help them.
On average each session with Erica lasted about 45 minutes.  So although this article is called 10 hours with Erica in reality I’ve probably only spent about 8 hours with her so far altogether.  I'm keeping a couple in reserve in case she ever needs them in the future.

One day as I saw a girl I didn’t know come running out the front door of my son’s primary school crying her heart out.   She sat down in the stone garden sobbing violently.  While my own children played I went and sat beside her with my arm around her.
After a little while her mum appeared from school and shouted for her.  They left together.
A week or so later I spotted the mother and went to chat to her to check that it had been okay that I’d put my arm around her daughter.  A very sad tale of how miserable Erica (her daughter) was in maths came tumbling out.   Erica had severe hearing difficulties until she was 7 and when she began to hear she had so much to catch up on she simply refused to engage with maths at all.  Now, in year 6 and with SATS approaching, her inability with maths was overwhelming her and dominating her life.  Mum was looking for a tutor.  I offered to tutor Erica if her mum would look after my children while I did so, an offer which was readily accepted.

Session 1: Exploring shape
I opened a box of ATM mats, which are regular polygons all with the same side length manufactured like beer mats.  I simply asked Erica to create patterns.  She created a tessellation, so we discussed tessellations and what they were.  Then she created a picture with mirror symmetry so we discussed and explored mirror symmetry.
We found that although she knew the names triangle and square Erica did not know pentagon, hexagon, octagon or decagon.  We learned those names by linking octagon to octopus, decagon to decimal and hex to six (both 3 letters end in x).
Each time Erica created a pattern I got her mum to come and look at it and to cherish and understand what she had done.   The atmosphere was happy and very positive.
Erica asked if she could draw her patterns so I got some wallpaper and she drew them by drawing round the shapes. 
For homework I asked her to learn the shape names and create more patterns on wallpaper – including at least one tessellation and one symmetrical picture.
Erica was highly literate and very clearly someone who would benefit from teaching which was always based around things she could see.  I explicitly reassured her and her mum that that was what she was going to get from me and that it would overcome her problems. 

Session 2: Ways of adding and base 10.
Erica had completed her homework beautifully and had made some observations about which shapes tessellate and which don’t which she was keen to share.  We put her drawings away so that we would be able to come back to them as an introduction to talking about angle at some time in the future.
I gave Erica some piles of blocks to count and after letting her do that for a while we explored how building them into sticks of 10 could help her ‘see’ how many there were more easily.
I then gave her two piles of blocks at a time to add.  Erica was not comfortable working in base 10.  She wanted to count all and had a preference for grouping the blocks into fives to help her do that.  I let her do that for a while and then showed her how grouping the blocks into 10s (as earlier) gave us a way of finding the sum more rapidly.  We took time to talk about the difference between her preferred method and the method her teacher would have been trying to teach her and to show her that they weren’t very different.  I reassured her that by using visual aids such as multilink blocks she would be able to communicate much more easily with me and her teacher and that would help her teacher understand her methods and her understand her teachers methods much more easily than had happened before.
For homework I set her some simple sums and asked her to use blocks to help her do them.  I asked her to tick the answers she was confident with, mark the ones she was not so sure with and write some sums of her own.

Session 3: Reviewing adding in base 10 and clip together shapes
We reviewed Erica’s sums.  She’d marked which she thought were right and which she was confused about.  We just explored them, using the base 10 blocks and puzzling out how to work through them.
Then we got out the clicksy polygons and made some 3D shapes, talking about the names of the flat shapes and starting to look at the properties of the solids.  I tried to talk about edges and corners but the way the bits fit together confused that as the parts of the edges which pop inside each other look like extra corners.
Homework was to build interesting solids from the clip together shapes.
Erica talked about how she’d been doing averages that day at school and hadn’t had a clue what was going on but it hadn’t mattered because her teacher had taught a ‘cheats method’ to help her find the answer with a calculator.
At this stage I started to try to find some teaching materials Erica could constructively be working with in class at times when she couldn’t follow the main lesson.

Session 4: exploring nets and solid shapes
Erica brought her shapes she’d made (mainly irregular).  I asked her to choose her favourite shape and she chose a double length triangular prism she’d made.   We talked about each shape. 
This week we used an exercise I’d already introduced in previous sessions, where I tap around the walls of the room with a stick to get Erica to count up and down (I would name some numbers and then stay silent while I tapped backwards and forwards to get her to count up and down).  When she was confident I would focusing make the exercise more challenging by moving into negative and large numbers and making difficult jumps like ‘down one from 10000’.
This time I tried tapping down for doubles but Erica really struggled with this change. 
Then we explored a box of solid shapes – describing them and naming them.  We talked about nets and made them on wallpaper by drawing round the faces of a square-based-pyramid, cutting it out and exploring the use of flaps. 
We talked about nets in general and opened out the triangular prism from earlier to turn it into a net in different ways.  Erica can clearly manipulate nets in her head.
Homework: By drawing round the faces of the 3D solid shapes, create nets for them. Name each net with the help of the lists of the names of the solid shapes.

Session 5: Introducing the open number line
Erica brought with her the beautiful nets of 3D shapes she had made.  I asked her to choose which was her favourite and talk about it.  She chose the ‘tent prison’ and we laughed about that – clearly contrasting the spellings prison and prism.  She spoke about each shape in turn.  I helped her notice that on prisms the faces which are not ends are all rectangles and on pyramids the faces which are not the base are triangles.
Then we started work on addition with an open number line.  We started with the first number and Erica added the second by breaking it into easy bits.  We used units, tens and hundreds generally but Erica did start to group them.  We moved on as soon as she was confident from 38 + 25 to 134+33 to 326 + 107 and so on.  We looked at adding 199 by adding 200 and taking off 1 – talking more about the strategy and the route than the specific number.
For homework I lent Erica a pizza game and asked her to play it with her family to help her build her confidence with fractions.
I’m working hard on telling Erica very silly answers as if they are right and checking she pulls me up on that which she is beginning to do.  The extent to which she will accept and agree with silly answer I tell her is startling but this is beginning to change with the help of a lot of good humour.  I’m still involving her mother in her progress, both to cherish the positive progress which is being made and to involve her and Erica together in understanding precisely what I am doing and why I am doing it.

Session 6
We explored the pizzas Erica had been playing with, looking at different ways of making a whole and some equivalencies of fractions.  In each case I wrote the sum that we had made with pizza parts on paper and ensured Erica could follow what it represented but had no expectation she could work in this abstract on her own yet.
I asked her how many halves in a whole and she could see it was 2 but for thirds and quarters and fifths she had to make them.  Then it clicked and she knew 1/10ths would be 10 and so on.  We linked that to the names of fractions (1/10ths are called 1/10ths because they come from splitting 1 into 10 parts).
We left the fractions and moved onto creating a 2 dimensional number wall (based on the number tapping exercise in session 4) with blocks.  We used the language three fives and so on – looking at the variety of ways of counting up the totals and in particular looking at why three fives are five threes (you just turn the block of cubes round) and so on. 
I discovered that Erica could count in 2s by rote but couldn’t really puzzle out 4 twos and so on, so I set her a homework of learning to say the two times table which is written out for her with squares to count for the answers.
Erica commented that she’s being forced to do SATS preparation every day and that she got 6 in one test but then today she only got 2.  Her mum says she isn’t sleeping because she’s stressed.  I think this is the right time to talk to school about letting Erica do different work in class and I’m going to suggest they allow her to work through the year 2 workbook as I think this will give her confidence in her ability to do maths she really understands in school while also highlighting points she still does not understand from KS1.

Sessions 7-9
At this point I was able to get Erica, her parents, her class teacher, her numeracy coordinator and her head teacher all to agree to take Erica off SATS preparation in class and to give her a year 2 workbook which she should work on both in class and with me.  In her sessions with me we talked ‘around topics’ – exploring them in wider contexts and with a clear focus on visual structures.  I found it useful to have the year 2 workbook to prompt me to explore gaps in Erica’s knowledge I might not otherwise have spotted.
This helped me identify that Erica’s had a lack of perceptual awareness of perpendicular lines and that she struggled to visualise clockwise vs anti-clockwise motion.  I’ve started to work on this with Erica focusing in particular on kinaesthetic activities but more work is needed.
As we introduced multiplication, I gave Erica visual times tables (3 fours are with a picture of 3 rows of four squares, 4 fours are with a picture of four rows of four squares and so on) which helped her properly ‘see multiplication’ as being repeated addition.
Erica rapidly completed the year 2 work book and we moved on to the year 4 resources.  When we introduced division I set it up as a splitting structure for her which she could confidently use – e.g. for 21 divided by 3 she would draw 3 people and share out 21 things between them.  But we also repeatedly discussed that there was also a chunking structure – drawing the matrix layout which links the two results.  They aim was that if a teacher used chunking or the reverse of multiplication to solve a division problem Erica would not be confused – she would simply know there are other methods and that she only needs to engage with them as far as she feels able because she can return to the structure she is confident with.
Changing Erica’s work in school led to a step-change in her engagement with and enjoyment of maths which her teachers noticed and responded to enthusiastically.   Her attitude to maths away from school also changed and she began to actively look for and obviously enjoy opportunities to calculate.

Published with the permission of Erica and her mum both of whom I would like to publicly thank.

My recommendations on how to intervene to help students who are experiencing severe difficulties with maths are available here.  They are particularly appropriate for students who have had a disrupted early education.


  1. This comment has been removed by a blog administrator.

  2. Rebecca, this is lovely. Thank you. I'm so glad you ran into Erica and were able to help her.

  3. Thanks Sue - I feel very lucky to have been able to work with Erica.

  4. This is one of the most brilliant pieces of teaching I've ever heard or read about. Congratulations. Definitely up there with Anne Sullivan's work with Helen Keller.

  5. John thank you for the compliment but it really is not due.

    I've only spent about 8 hours with Erica. Anne Sullivan spent many year living and working with Helen Keller.

    Anne Sullivan developed her methods for herself. Mine were mainly taught to me by my excellent PGCE tutors and friends and conferences session presenters at the Association of Teachers of Mathematics.